Engineering Calculus Notes 51

Engineering Calculus Notes 51 - b c Figure 1.23: Theorem...

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1.3. LINES IN SPACE 39 We will use these ideas to prove the following. Theorem 1.3.2. In any triangle, the three lines joining a vertex to the midpoint of the opposite side meet at a single point. Proof. Label the vertices of the triangle A , B and C , and their position vectors −→ a , −→ b and −→ c , respectively. Label the midpoint of each side with the name of the opposite vertex, primed; thus the midpoint of BC (the side opposite vertex A ) is A (see Figure 1.23 ). From Remark 1.3.1 we see x y z b b b A A B B C C A B C −→ a −→
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Unformatted text preview: b c Figure 1.23: Theorem 1.3.2 that the position vectors of the midpoints of the sides are O A = 1 2 ( b + c ) O B = 1 2 ( c + a ) O C = 1 2 ( a + b ) , and so the line A through A and A can be parametrized (using r as the parameter) by p A ( r ) = (1 r ) a + r 2 ( b + c ) = (1 r ) a + r 2 b + r 2 c ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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